To find the perimeters of the triangles, we need to sum the lengths of their sides.
Smaller Triangle:
- Side lengths: 6, 6, and 9 units.
- Perimeter = 6 + 6 + 9 = 21 units.
Larger Triangle:
- Side lengths: 12, 12, and 18 units.
- Perimeter = 12 + 12 + 18 = 42 units.
Now, we compute the ratio of the perimeter of the smaller triangle to the perimeter of the larger triangle:
\[ \text{Ratio} = \frac{\text{Perimeter of smaller triangle}}{\text{Perimeter of larger triangle}} = \frac{21}{42} = \frac{1}{2}. \]
Thus, the ratio of the perimeter of the smaller triangle to the perimeter of the larger triangle is:
\[ \boxed{\frac{1}{2}}. \]
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